Solvability of a class of boundary value problems for second-order operator-differential equations with a discontinuous coefficient in a weighted space (Q946111)

The paper is concerned with the differential-operator equation \[ \lambda u(x)-u''(x)+Au(x)=f(x),\quad x\in (0,1) \] and the boundary conditions \[ \lambda u(0)-\alpha u'(0)+T_1u=f_1, \] \[ \lambda u(1)+\beta u'(1)+T_2u=f_2, \] where \(\lambda\) is the spectral parameter, \(A\) is a self-adjoint, positive-definite operator in a Hilbert space \(H\) and \(T_k\) (\(k=1,2\)) are operators from \(L_p((0,1);H)\) into \(H\). First, the existence and estimates of the solutions are obtained. Next, the discreteness of the spectrum and the completeness of a system of root functions are established. Finally, an application of these abstract results to partial differential equations of second order is also given.